NAG CL Interfacef01hbc (complex_​gen_​matrix_​actexp_​rcomm)

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1Purpose

f01hbc computes the action of the matrix exponential ${e}^{tA}$, on the matrix $B$, where $A$ is a complex $n×n$ matrix, $B$ is a complex $n×m$ matrix and $t$ is a complex scalar. It uses reverse communication for evaluating matrix products, so that the matrix $A$ is not accessed explicitly.

2Specification

 #include
 void f01hbc (Integer *irevcm, Integer n, Integer m, Complex b[], Integer pdb, Complex t, Complex tr, Complex b2[], Integer pdb2, Complex x[], Integer pdx, Complex y[], Integer pdy, Complex p[], Complex r[], Complex z[], Complex ccomm[], double comm[], Integer icomm[], NagError *fail)
The function may be called by the names: f01hbc or nag_matop_complex_gen_matrix_actexp_rcomm.

3Description

${e}^{tA}B$ is computed using the algorithm described in Al–Mohy and Higham (2011) which uses a truncated Taylor series to compute the ${e}^{tA}B$ without explicitly forming ${e}^{tA}$.
The algorithm does not explicity need to access the elements of $A$; it only requires the result of matrix multiplications of the form $AX$ or ${A}^{\mathrm{H}}Y$. A reverse communication interface is used, in which control is returned to the calling program whenever a matrix product is required.

4References

Al–Mohy A H and Higham N J (2011) Computing the action of the matrix exponential, with an application to exponential integrators SIAM J. Sci. Statist. Comput. 33(2) 488-511
Higham N J (2008) Functions of Matrices: Theory and Computation SIAM, Philadelphia, PA, USA

5Arguments

Note:  this function uses reverse communication. Its use involves an initial entry, intermediate exits and re-entries, and a final exit, as indicated by the argument irevcm. Between intermediate exits and re-entries, all arguments other than b2, x, y, p and r must remain unchanged.
1: $\mathbf{irevcm}$Integer * Input/Output
On initial entry: must be set to $0$.
On intermediate exit: ${\mathbf{irevcm}}=1$, $2$, $3$, $4$ or $5$. The calling program must:
1. (a)if ${\mathbf{irevcm}}=1$: evaluate ${B}_{2}=AB$, where ${B}_{2}$ is an $n×m$ matrix, and store the result in b2;
if ${\mathbf{irevcm}}=2$: evaluate $Y=AX$, where $X$ and $Y$ are $n×2$ matrices, and store the result in y;
if ${\mathbf{irevcm}}=3$: evaluate $X={A}^{\mathrm{H}}Y$ and store the result in x;
if ${\mathbf{irevcm}}=4$: evaluate $p=Az$ and store the result in p;
if ${\mathbf{irevcm}}=5$: evaluate $r={A}^{\mathrm{H}}z$ and store the result in r.
2. (b)call f01hbc again with all other parameters unchanged.
On final exit: ${\mathbf{irevcm}}=0$.
Note: any values you return to f01hbc as part of the reverse communication procedure should not include floating-point NaN (Not a Number) or infinity values, since these are not handled by f01hbc. If your code inadvertently does return any NaNs or infinities, f01hbc is likely to produce unexpected results.
2: $\mathbf{n}$Integer Input
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
3: $\mathbf{m}$Integer Input
On entry: the number of columns of the matrix $B$.
Constraint: ${\mathbf{m}}\ge 0$.
4: $\mathbf{b}\left[\mathit{dim}\right]$Complex Input/Output
Note: the dimension, dim, of the array b must be at least ${\mathbf{pdb}}×{\mathbf{m}}$.
The $\left(i,j\right)$th element of the matrix $B$ is stored in ${\mathbf{b}}\left[\left(j-1\right)×{\mathbf{pdb}}+i-1\right]$.
On initial entry: the $n×m$ matrix $B$.
On intermediate exit: if ${\mathbf{irevcm}}=1$, contains the $n×m$ matrix $B$.
On intermediate re-entry: must not be changed.
On final exit: the $n×m$ matrix ${e}^{tA}B$.
5: $\mathbf{pdb}$Integer Input
On entry: the stride separating matrix row elements in the array b.
Constraint: ${\mathbf{pdb}}\ge {\mathbf{n}}$.
6: $\mathbf{t}$Complex Input
On entry: the scalar $t$.
7: $\mathbf{tr}$Complex Input
On entry: the trace of $A$. If this is not available then any number can be supplied ($0.0$ is a reasonable default); however, in the trivial case, $n=1$, the result ${e}^{{\mathbf{tr}}t}B$ is immediately returned in the first row of $B$. See Section 9.
8: $\mathbf{b2}\left[\mathit{dim}\right]$Complex Input/Output
Note: the dimension, dim, of the array b2 must be at least ${\mathbf{pdb2}}×{\mathbf{m}}$.
The $\left(i,j\right)$th element of the matrix is stored in ${\mathbf{b2}}\left[\left(j-1\right)×{\mathbf{pdb2}}+i-1\right]$.
On initial entry: need not be set.
On intermediate re-entry: if ${\mathbf{irevcm}}=1$, must contain $AB$.
On final exit: the array is undefined.
9: $\mathbf{pdb2}$Integer Input
On entry: the stride separating matrix row elements in the array b2.
Constraint: ${\mathbf{pdb2}}\ge {\mathbf{n}}$.
10: $\mathbf{x}\left[\mathit{dim}\right]$Complex Input/Output
Note: the dimension, dim, of the array x must be at least ${\mathbf{pdx}}×2$.
The $\left(i,j\right)$th element of the matrix $X$ is stored in ${\mathbf{x}}\left[\left(j-1\right)×{\mathbf{pdx}}+i-1\right]$.
On initial entry: need not be set.
On intermediate exit: if ${\mathbf{irevcm}}=2$, contains the current $n×2$ matrix $X$.
On intermediate re-entry: if ${\mathbf{irevcm}}=3$, must contain ${A}^{\mathrm{H}}Y$.
On final exit: the array is undefined.
11: $\mathbf{pdx}$Integer Input
On entry: the stride separating matrix row elements in the array x.
Constraint: ${\mathbf{pdx}}\ge {\mathbf{n}}$.
12: $\mathbf{y}\left[\mathit{dim}\right]$Complex Input/Output
Note: the dimension, dim, of the array y must be at least ${\mathbf{pdy}}×2$.
The $\left(i,j\right)$th element of the matrix $Y$ is stored in ${\mathbf{y}}\left[\left(j-1\right)×{\mathbf{pdy}}+i-1\right]$.
On initial entry: need not be set.
On intermediate exit: if ${\mathbf{irevcm}}=3$, contains the current $n×2$ matrix $Y$.
On intermediate re-entry: if ${\mathbf{irevcm}}=2$, must contain $AX$.
On final exit: the array is undefined.
13: $\mathbf{pdy}$Integer Input
On entry: the stride separating matrix row elements in the array y.
Constraint: ${\mathbf{pdy}}\ge {\mathbf{n}}$.
14: $\mathbf{p}\left[{\mathbf{n}}\right]$Complex Input/Output
On initial entry: need not be set.
On intermediate re-entry: if ${\mathbf{irevcm}}=4$, must contain $Az$.
On final exit: the array is undefined.
15: $\mathbf{r}\left[{\mathbf{n}}\right]$Complex Input/Output
On initial entry: need not be set.
On intermediate re-entry: if ${\mathbf{irevcm}}=5$, must contain ${A}^{\mathrm{H}}z$.
On final exit: the array is undefined.
16: $\mathbf{z}\left[{\mathbf{n}}\right]$Complex Input/Output
On initial entry: need not be set.
On intermediate exit: if ${\mathbf{irevcm}}=4$ or $5$, contains the vector $z$.
On intermediate re-entry: must not be changed.
On final exit: the array is undefined.
17: $\mathbf{ccomm}\left[{\mathbf{n}}×\left({\mathbf{m}}+2\right)\right]$Complex Communication Array
18: $\mathbf{comm}\left[3×{\mathbf{n}}+14\right]$double Communication Array
19: $\mathbf{icomm}\left[2×{\mathbf{n}}+40\right]$Integer Communication Array
20: $\mathbf{fail}$NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
On entry, argument $⟨\mathit{\text{value}}⟩$ had an illegal value.
NE_INT
On entry, ${\mathbf{m}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{m}}\ge 0$.
On entry, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{n}}\ge 0$.
On initial entry, ${\mathbf{irevcm}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{irevcm}}=0$.
On intermediate re-entry, ${\mathbf{irevcm}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{irevcm}}=1$, $2$, $3$, $4$ or $5$.
NE_INT_2
On entry, ${\mathbf{pdb}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pdb}}\ge {\mathbf{n}}$.
On entry, ${\mathbf{pdb2}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pdb2}}\ge {\mathbf{n}}$.
On entry, ${\mathbf{pdx}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pdx}}\ge {\mathbf{n}}$.
On entry, ${\mathbf{pdy}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pdy}}\ge {\mathbf{n}}$.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NW_SOME_PRECISION_LOSS
${e}^{tA}B$ has been computed using an IEEE double precision Taylor series, although the arithmetic precision is higher than IEEE double precision.

7Accuracy

For an Hermitian matrix $A$ (for which ${A}^{\mathrm{H}}=A$) the computed matrix ${e}^{tA}B$ is guaranteed to be close to the exact matrix, that is, the method is forward stable. No such guarantee can be given for non-Hermitian matrices. See Section 4 of Al–Mohy and Higham (2011) for details and further discussion.

8Parallelism and Performance

f01hbc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9.1Use of $\mathbit{T}\mathbit{r}\left(\mathbit{A}\right)$

The elements of $A$ are not explicitly required by f01hbc. However, the trace of $A$ is used in the preprocessing phase of the algorithm. If $Tr\left(A\right)$ is not available to the calling function then any number can be supplied ($0$ is recommended). This will not affect the stability of the algorithm, but it may reduce its efficiency.

9.2When to use f01hbc

f01hbc is designed to be used when $A$ is large and sparse. Whenever a matrix multiplication is required, the function will return control to the calling program so that the multiplication can be done in the most efficient way possible. Note that ${e}^{tA}B$ will not, in general, be sparse even if $A$ is sparse.
If $A$ is small and dense then f01hac can be used to compute ${e}^{tA}B$ without the use of a reverse communication interface.
The real analog of f01hbc is f01gbc.

9.3Use in Conjunction with NAG Library Functions

To compute ${e}^{tA}B$, the following skeleton code can normally be used:
```do {
f01hbc(&irevcm,n,m,b,tdb,t,tr,b2,tdb2,x,tdx,y,tdy,p,r,z,ccomm,comm,  &
icomm,&fail);
if (irevcm == 1) {
.. Code to compute B2=AB ..
}
else if (irevcm == 2){
.. Code to compute Y=AX ..
}
else if (irevcm == 3){
.. Code to compute X=A^H Y ..
}
else if (irevcm == 4){
.. Code to compute P=AZ ..
}
else if (irevcm == 5){
.. Code to compute R=A^H Z ..
}
} (while irevcm !=0)```
The code used to compute the matrix products will vary depending on the way $A$ is stored. If all the elements of $A$ are stored explicitly, then f16zac can be used. If $A$ is triangular then f16zfc should be used. If $A$ is Hermitian, then f16zcc should be used. If $A$ is symmetric, then f16ztc should be used. For sparse $A$ stored in coordinate storage format f11xnc and f11xsc can be used.

10Example

This example computes ${e}^{tA}B$ where
 $A = ( 0.7+0.8i -0.2+0.0i 1.0+0.0i 0.6+0.5i 0.3+0.7i 0.7+0.0i 0.9+3.0i 1.0+0.8i 0.3+3.0i -0.7+0.0i 0.2+0.6i 0.7+0.5i 0.0+0.9i 4.0+0.0i 0.0+0.0i 0.2+0.0i ) ,$
 $B = ( 0.1+0.0i 1.2+0.1i 1.3+0.9i -0.2+2.0i 4.0+0.6i -1.0+0.8i 0.4+0.0i -0.9+0.0i )$
and
 $t=1.1+0.0i .$
$A$ is stored in compressed column storage format (CCS) and matrix multiplications are performed using the function matmul.

10.1Program Text

Program Text (f01hbce.c)

10.2Program Data

Program Data (f01hbce.d)

10.3Program Results

Program Results (f01hbce.r)